Please do not say that you mean one thing by "infinity" and then change in your response!

When asked, "What do you mean by infinity", you responded "its a limit" (which is pretty much meaningless) a dx responded to that with "If you mean [itex]\lim_{x\to \infty} x- x[/itex] then it is 0".

He did NOT say "infinity- infinity = 0". He was trying to respond to your vague answer.

He could as well have pointed out that [itex]\lim_{x\to \infty} x^2- x[/itex] is also "infinity minus infinity", in that [itex]lim_{x\to \infty}x^2= \infty[/itex] and [itex]\lim_{x\to \infty} x= \infty[/itex], and that limit is equal to infinity. In fact, given any number a, [itex]\lim_{x\to \infty} x+ a= \infty[/itex] and [itex]\lim_{x\to \infty}= \infty[/itex] so [itex]\lim_{x\to\infty}(x+a)- x[/itex] can be said to be "infinity - infinity" but that limit is obviously a. If, by "infinity" you mean "its a limit" then, depending on exactly which limit you use you can make "infinity - infinity" equal to anything.

What you need to understand is that when we talk about "[itex]\lim_{x\rightarrow \infty} f(x)[/itex] or [itex]\lim_{n\rightarrow\infty} a_n[/itex], that "infinity" is just short hand for "x (or n) increases without bound". Also saying that [itex]\lim_{x\rightarrow a} f(x)= \infty[/itex] or [itex]\lim_{n\rightarrow \infty} a_n= \infty[/itex] we are NOT saying that the limit is "the number infinity", we are saying that the limit does not exist in a particular way.

In many text books they will say, for example, that [itex]\lim_{x\to a} x^2[/itex] converges to [itex]a^2[/itex] but that [itex]\lim_{x\to 0} 1/x[/itex] diverges to infinity- that is, the limit does not exist.

but back to jontyjashan, the simple answer is that infinity minus infinity can not be defined, because it can be anything. it's like asking, what is anything divided by zero? it doesn't make sense to ask a question like that.